White Shark Number of Breeders Re-Analysis 2024

Parameter Choices

#use the fixed cohorts
new_cohort_assignment <- TRUE

#use the new fixed genomes
use_fixed_genotypes <- TRUE

#adjust these if you want the relevant chunks to run
rerun_COLONY <- FALSE
  rerun_COLONY_all <- FALSE
  
rerun_LDNE <- TRUE

include_2014 <- TRUE #will still run 2014 estimates, just not included in the analysis

#p_crit choices
p_crit_list <- c(0, 0.02, 0.05) 
p_crit_chosen <- 0.05

#COLONY params for simulations that will actually be used
COLONY_run_length <- 2
COLONY_allele_freq <- NULL # vs "overall" for population averages
COLONY_dropout <- 0 
COLONY_error_rate <- 0.02
COLONY_random_seed <- 1

#using old data for cohorts and genotypes, no 2014, pcrit 2014, defaults for COLONY should regenerate the original published results exactly

Overview

This document/repository contains and outlines the re-analysis of the data presented in Davenport et al 2021 Davenport et al. (2021), as re-re-analysed by Andrew Jones.

The authors were made aware of a bug or error in the output-files used in both NeEstimator and Colony to estimate the number of breeders per cohort. This error caused the data to become shuffled. Below, analysis is presented comparing the published data to files made directly from the original raw data. The input files for Nb estimation for NeEstimator and COLONY are re-made using raw data (based on the published loci) and the updated results are presented. The published work further sought to combine estimators of effective size to make reporting and decision making easier - combined estimates of Nb per-cohort. Here using the revised results we also make these re-estimates.

An additional sensitivity analysis is performed on the full dataset to illustrate how the inclusion of different SNPs and individuals can affect final point-estimates of effective size, and highlights the importance of considering confidence intervals (i.e using a 95%CI we expect that 95% of the time the true value of the parameter lies in its interval), as well as various estimators in conservation decisions. S

Background

In Davenport et al., 2021, the effective number of breeders (Nb) was estimated in 4 consecutive cohorts (2010-2013) based on the premise that when a genetic sample contains only individuals from a single age cohort (a group of individuals having the same age-class), then the estimate of effective population size (Ne) from a single-sample estimator in species with overlapping generations corresponds to the effective number of breeders (Nb). For long-lived, iteroparous species such as the white shark, estimates of Nb are generally considered more useful for monitoring as they apply to a single breeding season (rather than needing a sample representative of a generation - an assumption of Ne (see Waples (2022))), and it represents an accessible parameter for monitoring population trends at ecological timescales most relevant to conservation and management needs. Nb is particularly useful in cases where juvenile or YOY samples can be easily collected (i.e. through SMART shark program NSW). Estimating Nb and monitoring its change over time allows timely detection of population trends (decline, restoration, recovery, expansion), even if using as few as four or five consecutive reproductive cycles (Leberg (2005); Wang (2005); Luikart et al. (2021)).

Data

The original DArT data for this project can be found in the file /Data/Raw

The results of the reanalysis can be found in /Results.

The filtered list of loci from the published analysis can be found in Data/Raw/Report-DSha18-3402/File_Formats/Genepop_NeEstimator_Input/nswdpi_whiteshark_gp.gen.

The loci names were extracted from this document to ensure the data from the publication are used in the reanalysis here. Readers are referred to the original publication Davenport et al. (2021) for more detailed methods. In summary, a list of loci/SNPs which meet the articles filtering criteria are saved in the ‘.gen’ file. These are used here to re-make the input files for NeEstimator and COLONY from the original raw-data, from which Nb per cohort is re-estimated.The full raw data is also used to re-estimate Nb using a sensitivity analysis of loci/individuals.

The processed Data is found in /Data/processed.

Confirmation of Issue in Original Manuscript

For this section I follow the approach used by Dani as far as was required to confirm the work of Dean and Paul.

Setup data

A bug in an R-package used in the original publication is suspected to have introduced inconsistencies into the output files used in the final analysis.

# read in published strata
strata_info = read.table("Original Project Files/Strata_Sex_Model_COHORT_REGRESSION.tsv", header = T) # used in the published ms 

# read in published genepop file affected by bug - herein refered to as 'orig-genepop'
orig_gp <- adegenet::read.genepop(file = "Original Project Files/File_Formats/Genepop_NeEstimator_Input/nswdpi_whiteshark_gp.gen", ncode = 3L, quiet = TRUE)
orig_gl <- dartR::gi2gl(orig_gp)

## Starting :: 
##  Starting dartR 
##  Starting gi2gl 
## Starting gl.compliance.check 
##   Processing genlight object with SNP data
##   The slot loc.all, which stores allele name for each locus, is empty. 
## Creating a dummy variable (A/C) to insert in this slot. 
##   Checking coding of SNPs
##     SNP data scored NA, 0, 1 or 2 confirmed
##   Checking locus metrics and flags
##   Recalculating locus metrics
##   Checking for monomorphic loci
##     Dataset contains monomorphic loci
##   Checking for loci with all missing data
##     No loci with all missing data detected
##   Checking whether individual names are unique.
##   Checking for individual metrics
##   Warning: Creating a slot for individual metrics
##   Checking for population assignments
##     Population assignments confirmed
##   Spelling of coordinates checked and changed if necessary to 
##             lat/lon
## Completed: gl.compliance.check 
## Completed: :: 
##  Completed: dartR 
##  Completed: gi2gl

# it didnt fix the names, so do it here
orig_gl@loc.names<- sub(".*?-", "", str_replace_all(orig_gl@loc.names,"__","-"))  # change whitelist loci seperators so they are in the fornat needed to match the genlight obj

# 'new_genepop'
# read in orgional data, and use the list of loci to make a new dataset - herein refered to as 'new-genepop'
input_gl = dartR::gl.read.dart("Data/Raw/Report_DSha18-3402_SNP_2_ReLabeled.csv")

## Starting :: 
##  Starting dartR 
##  Starting gl.read.dart 
## Starting utils.read.dart 
##   Topskip not provided.
##  Setting topskip to 6 .
##   Reading in the SNP data
##   Detected 2 row format.
## Number of rows per clone (should be only  2 s): 2 
##  Added the following locus metrics:
## AlleleID AlleleSequence TrimmedSequence Chrom_Shark_Callorhinchus_v1 ChromPos_Shark_Callorhinchus_v1 AlnCnt_Shark_Callorhinchus_v1 AlnEvalue_Shark_Callorhinchus_v1 SNP SnpPosition CallRate OneRatioRef OneRatioSnp FreqHomRef FreqHomSnp FreqHets PICRef PICSnp AvgPIC AvgCountRef AvgCountSnp RepAvg .
## Recognised: 278 individuals and 9841  SNPs in a 2 row format using Data/Raw/Report_DSha18-3402_SNP_2_ReLabeled.csv 
## Completed: utils.read.dart 
## Starting utils.dart2genlight 
## Starting conversion....
## Format is 2 rows.
## Please note conversion of bigger data sets will take some time!
## Once finished, we recommend to save the object using save(object, file="object.rdata")
## Completed: utils.dart2genlight 
##  Data read in. Please check carefully the output above
##   Read depth calculated and added to the locus metrics
##   Minor Allele Frequency (MAF) calculated and added to the locus metrics
##   Recalculating locus metrics provided by DArT (optionally specified)
## Completed: :: 
##  Completed: dartR 
##  Completed: gl.read.dart

# change the GL loci names so they are in a suitable format 
input_gl@loc.names <- sub("^([^-]*-[^-]*).*", "\\1", input_gl@loc.names)
# filter the gl to have the same loci as used in 'orig-gp'
order_loci <- sapply(orig_gl@loc.names, function(x,order){which(order == x)}, order = input_gl@loc.names)
new_gl <- input_gl[,order_loci]
# dartR has a bug where it doesnt fix up other information stored in other
dim(new_gl@other$loc.metrics)

## [1] 3668   24

new_gl@other$loc.metrics <- new_gl@other$loc.metrics[order_loci,]
# match so the files have the same samples, in the same order
order_ind <- sapply(orig_gl@ind.names, function(x,order){which(order == x)}, order = str_replace_all(new_gl@ind.names, "_", "-"))

new_gl <- new_gl[order_ind,]
# check
dim(new_gl@other$loc.metrics)

## [1] 3668   24

# add the population information, also make sure its in the right order
pop = dplyr::left_join(tibble::tibble(TARGET_ID=str_replace_all(new_gl@ind.names, "_", "-")), strata_info, by = "TARGET_ID")
new_gl@pop<- factor(pop$STRATA)
# check 
which(head(str_replace_all(new_gl@ind.names, "_", "-")) ==  head(orig_gl@ind.names))

## [1] 1 2 3 4 5 6

which(head(orig_gl@loc.names) == head(new_gl@loc.names))

## [1] 1 2 3 4 5 6

Quantify differences in missingness between the published and new dataset

# compare the old and the new files for missingness
# print out each genlight to get info on missngness, loci, invidual
orig_gl

##  /// GENLIGHT OBJECT /////////
## 
##  // 241 genotypes,  3,668 binary SNPs, size: 1.2 Mb
##  4506 (0.51 %) missing data
## 
##  // Basic content
##    @gen: list of 241 SNPbin
##    @ploidy: ploidy of each individual  (range: 2-2)
## 
##  // Optional content
##    @ind.names:  241 individual labels
##    @loc.names:  3668 locus labels
##    @loc.all:  3668 alleles
##    @pop: population of each individual (group size range: 1-63)
##    @other: a list containing: loc.metrics  loc.metrics.flags  verbose  history  ind.metrics

# compare the old and the new files for missingness
# print out each genlight to get info on missngness, loci, invidual
new_gl

##  ********************
##  *** DARTR OBJECT ***
##  ********************
## 
##  ** 241 genotypes,  3,668 SNPs , size: 5.7 Mb
## 
##     missing data: 4503 (=0.51 %) scored as NA
## 
##  ** Genetic data
##    @gen: list of 241 SNPbin
##    @ploidy: ploidy of each individual  (range: 2-2)
## 
##  ** Additional data
##    @ind.names:  241 individual labels
##    @loc.names:  3668 locus labels
##    @loc.all:  3668 allele labels
##    @position: integer storing positions of the SNPs [within 69 base sequence]
##    @pop: population of each individual (group size range: 1-63)
##    @other: a list containing: loc.metrics, ind.metrics, loc.metrics.flags, verbose, history 
##     @other$ind.metrics: service, plate_location 
##     @other$loc.metrics: AlleleID, AlleleSequence, TrimmedSequence, Chrom_Shark_Callorhinchus_v1, ChromPos_Shark_Callorhinchus_v1, AlnCnt_Shark_Callorhinchus_v1, AlnEvalue_Shark_Callorhinchus_v1, SNP, SnpPosition, CallRate, OneRatioRef, OneRatioSnp, FreqHomRef, FreqHomSnp, FreqHets, PICRef, PICSnp, AvgPIC, AvgCountRef, AvgCountSnp, RepAvg, clone, uid, rdepth 
##    @other$latlon[g]: no coordinates attached

These results show the number of individuals and the number of loci does not change between the published and the new dataset. However, the number of missing data in the published dataset is 4506, while the re-analysis dataset missingingess is 4503.

Visualise the difference between the published and new dataset

# visualize the data
ogplot = adegenet::glPlot(orig_gl, posi="topleft")

newplot = adegenet::glPlot(new_gl, posi="topleft")

Discussion of error

More details available in Dani’s report and Dean and Paul’s report.

Main points:

• Error definitely present as described in analyis by Dean and Paul.
• Error changes data in a way that would be expected to have some impact estimates of Nb
• Error needs to be corrected

Corrected Ne Estimates

New cohort assignment

The new cohort assignment file “Data/Processed/New_Cohort_Assignment.tsv” was created by running the script Scripts/Age_at_Length.R.

The von Bertanlaffy parameters used are from are from O’Connor (2011).

meta_data <- readr::read_csv("Data/Raw/Copy of UPDTAED LAT LONGS UQ genetics PhD181018 Paul Butcher NSW DPI.csv") %>%
  #select(MBB_Code, FL, TL) %>%
  mutate(date_column = as.Date(`Date tagged`, format="%d/%m/%y"),
         modified_year = year(date_column))

# 1. relationship between TL and FL 
lm1 <- lm(meta_data$TL ~ meta_data$FL)

Tl_intercept <- lm1$coefficients[1]
Tl_slope <- lm1$coefficients[2]
#
Tl_intercept <- floor(Tl_intercept * 100) / 100
Tl_slope <- floor(Tl_slope * 100) / 100 # eq (1) manuscript , round down to two decimal places

TL <- sapply(meta_data$FL, function(x) {Tl_intercept + x * Tl_slope}) # eq (1) manuscript 

# Estimate age directly using values from O'Connor 2011 and FL
#solve for age
vbgf3 <- function(L, L_infinity, K_value, t_0) {
  result <- t_0 - (1 / K_value) * log(1 - (L / L_infinity))
  return(result)
}
# Estimating age at length
# info:
# Best fitting parmaters for Males
Linf <- 798.94# cm TL
k <- 0.047 # 
L0 <- 140 #cm 
T0 <- -3.8 #years 
Males <- list(L_infinity_value = Linf, K_value = k, L0 = L0, t_0_value = T0)

# Best fitting paramters for Females 
Linf<- 719.02# cm TL
k<- 0.056 # 
L0<- 140 #cm 
T0 <- -3.8 #years 
Females <- list(L_infinity_value = Linf, K_value = k, L0 = L0, t_0_value = T0)

Linf<- 746.66# cm TL
k<- 0.053# 
L0<- 140 #cm 
T0 <- -3.8 #years 
Combined <- list(L_infinity_value = Linf, K_value = k, L0 = L0, t_0_value = T0)

# female 
idx_f<- which(meta_data$Sex == "F")
idx_m<- which(meta_data$Sex == "M")

male_t = sapply(TL[idx_m], function(x) {vbgf3(x, Males[["L_infinity_value"]], Males[["K_value"]], Males[["t_0_value"]])})
female_t = sapply(TL[idx_f], function(x) {vbgf3(x, Females[["L_infinity_value"]], Females[["K_value"]], Females[["t_0_value"]])})

# Cohort is equal to year sampled minus age
male_age = trunc(meta_data$modified_year[idx_m] -  male_t)
female_age = trunc(meta_data$modified_year[idx_f] -  female_t) # including some samples, ie. MBB1348 depends on how you round the age or the year
output = tibble::tibble(STRATA = c(male_age, female_age), MBB_CODE = c(meta_data$MBB_Code[idx_m], meta_data$MBB_Code[idx_f]))
readr::write_tsv(output, "Data/Processed/New_Cohort_Assignment.tsv")

New Input files for NeEstimator and COLONY

Below the input files for NeEstimator and COLONY are remade to make new estimates of Nb using a new file made from the published list of loci (“new_gp”).

dartR is used write out a genepop file, and a custom script to write out a COLONY file.

# 1. write out new_gp as a genepop , check that the file has been written out correctly 
# 1a. run this file through NeEstimator

# first its important to make sure that the cohort are correctly assigned 

#some sharks no cohort - why ?no meta data?

if(new_cohort_assignment){
  cohort_assignment_file <- "Data/Processed/New_Cohort_Assignment.tsv"
  
  cohort_assignment = readr::read_tsv(cohort_assignment_file) %>% #New cohort assignment
  separate(MBB_CODE, into = c("string1", "string2"), sep = " ", remove = FALSE) %>%
  unite(MBB_CODE_JOIN, string1, string2, sep = "_") 
  
}else{
  cohort_assignment_file <- "Original Project Files/Strata_Sex_Model_COHORT_REGRESSION.tsv"
  
  cohort_assignment = readr::read_tsv(cohort_assignment_file) %>% #New cohort assignment
  separate(INDIVIDUALS, into = c("string1", "string2"), sep = "-", remove = FALSE) %>%
  unite(MBB_CODE_JOIN, string1, string2, sep = "_") 
}



if(use_fixed_genotypes){
  gl_file <- new_gl
}else{
  gl_file <- orig_gl
}

gl_file$ind.names <- str_replace(gl_file$ind.names, "-", "_")


ind_order = tibble::tibble(MBB_CODE_JOIN = gl_file$ind.names, OLD_STRATA = gl_file@pop) %>%
  left_join(., cohort_assignment, by = "MBB_CODE_JOIN")

gl_file@pop <- as.factor(ind_order$STRATA)
table(gl_file@pop)

## 
## 2005 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 
##    1    3    6   11   29   41   52   64   23    9    2

dartR::gl2genepop(gl_file, outfile = "genotypes_for_Ne.gen", outpath = "Data/Processed/NeEstimator")

## Starting :: 
##  Starting dartR 
##  Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  Data/Processed/NeEstimator/genotypes_for_Ne.gen/
## Completed: :: 
##  Completed: dartR 
##  Completed: gl2genepop

Run COLONY

#need to read COLONY data in from files

colony_results_files <- list.files("./Results/COLONY/")

col_list <- list()

for (i in seq_len(length(colony_results_files))){
  c_r_f <- colony_results_files[i]
  res <- readLines(file.path("./Results/COLONY/", c_r_f))
  temp <- str_match(c_r_f, "^COLONY_output_(scenario_[a-zA-Z0-9]+)_([0-9]{4}).txt$")
  
  Year <- as.numeric(temp[3])
  Scenario <- temp[2]
  Nb_SA_est <- as.numeric(na.omit(str_match(res, "^Ne\\s+=\\s+([0-9]+)$")[, 2])[1]) 
  Nb_SA_upper <- as.numeric(na.omit(str_match(res, "^CI95\\(U\\)\\s+=\\s+([0-9]+)$")[, 2])[1]) 
  Nb_SA_lower <- as.numeric(na.omit(str_match(res, "^CI95\\(L\\)\\s+=\\s+([0-9]+)$")[, 2])[1]) 
  
  col_list[[i]] <- data.frame(Year, Scenario, Nb_SA_est, Nb_SA_lower, Nb_SA_upper)
}

All_COLONY_data <- bind_rows(col_list) %>% filter(Year >= 2010, Year <= 2014)

write_csv(All_COLONY_data, "./Results/COLONY/ResultsSummary.csv")

Scenario List:

• scenario_00 - Default settings, short run
• scenario_01 - Default settings, population allele frequencies, short run
• scenario_02 - Error rates from Wang 2018, short run
• scenario_03 - Error rates from Wang 2018, population allele frequencies, short run
• scenario_04 - Error rates inbetween 1, short run
• scenario_05 - Error rates inbetween 1, population allele frequencies, short run
• scenario_06 - Error rates inbetween 2, short run
• scenario_07 - Error rates inbetween 2, population allele frequencies, short run
• scenario_08 - Default settings, long run, repeat - Settings from original paper
• scenario_09 - Default settings, long run, repeat
• scenario_10 - Default settings, medium run, repeat
• scenario_11 - Default settings, medium run, repeat

ggplot(All_COLONY_data, aes(x = Year, y = Nb_SA_est, colour = Scenario, group=Scenario)) + geom_point(position =dodge) + geom_errorbar(aes(ymin = Nb_SA_lower, ymax = Nb_SA_upper), width = width, position =dodge) 

All_COLONY_data %>% filter(Scenario %in% c("scenario_00", "scenario_01", "scenario_02", "scenario_03","scenario_08", "scenario_final")) %>% ggplot( aes(x = Year, y = Nb_SA_est, colour = Scenario, group=Scenario)) + geom_point(position =dodge) + geom_errorbar(aes(ymin = Nb_SA_lower, ymax = Nb_SA_upper), width = width, position =dodge) 

Nb_SA_data <- All_COLONY_data %>% filter(Scenario == "scenario_final") #need to pick one(1) for example scenario_00 is the defaults with 'quick run', scenario_08 is the original settings
write_csv(Nb_SA_data, "./Results/COLONY/ResultsSummary_final.csv")

Estimates slightly lower for some years if use population allele frequencies, otherwise basically same. No real reason to deviate from original settings.

Run NeEstimator

## Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  /tmp/RtmpSfUVyo/dummy.gen/
## Completed: gl2genepop 
## $`2010_MBB`
##                     Statistic Frequency 1
##  Lowest Allele Frequency Used          0+
##     Harmonic Mean Sample Size        28.8
##       Independent Comparisons     4639447
##                   OverAll r^2    0.041995
##           Expected r^2 Sample     0.03859
##                 Estimated Ne^        88.7
##             CI low Parametric        87.3
##            CI high Parametric        90.2
##              CI low JackKnife        48.5
##             CI high JackKnife       332.6
## 
## Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  /tmp/RtmpSfUVyo/dummy.gen/
## Completed: gl2genepop 
## $`2011_MBB`
##                     Statistic Frequency 1
##  Lowest Allele Frequency Used          0+
##     Harmonic Mean Sample Size        40.4
##       Independent Comparisons     5352795
##                   OverAll r^2    0.027775
##           Expected r^2 Sample    0.026659
##                 Estimated Ne^       296.6
##             CI low Parametric         288
##            CI high Parametric       305.8
##              CI low JackKnife       160.1
##             CI high JackKnife      1498.1
## 
## Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  /tmp/RtmpSfUVyo/dummy.gen/
## Completed: gl2genepop 
## $`2012_MBB`
##                     Statistic Frequency 1
##  Lowest Allele Frequency Used          0+
##     Harmonic Mean Sample Size        51.5
##       Independent Comparisons     5663913
##                   OverAll r^2    0.021786
##           Expected r^2 Sample    0.020592
##                 Estimated Ne^       277.2
##             CI low Parametric       271.3
##            CI high Parametric       283.2
##              CI low JackKnife       166.7
##             CI high JackKnife         735
## 
## Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  /tmp/RtmpSfUVyo/dummy.gen/
## Completed: gl2genepop 
## $`2013_MBB`
##                     Statistic Frequency 1
##  Lowest Allele Frequency Used          0+
##     Harmonic Mean Sample Size        63.2
##       Independent Comparisons     6045258
##                   OverAll r^2    0.018056
##           Expected r^2 Sample    0.016599
##                 Estimated Ne^       226.6
##             CI low Parametric       223.5
##            CI high Parametric       229.9
##              CI low JackKnife       153.3
##             CI high JackKnife         411
## 
## Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  /tmp/RtmpSfUVyo/dummy.gen/
## Completed: gl2genepop 
## $`2014_MBB`
##                     Statistic Frequency 1
##  Lowest Allele Frequency Used          0+
##     Harmonic Mean Sample Size        22.9
##       Independent Comparisons     4454940
##                   OverAll r^2    0.051739
##           Expected r^2 Sample     0.04988
##                 Estimated Ne^       163.9
##             CI low Parametric       158.1
##            CI high Parametric       170.2
##              CI low JackKnife        75.9
##             CI high JackKnife         Inf
## 
## Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  /tmp/RtmpSfUVyo/dummy.gen/
## Completed: gl2genepop 
## $`2010_MBB`
##                     Statistic Frequency 1 Frequency 2
##  Lowest Allele Frequency Used       0.020          0+
##     Harmonic Mean Sample Size        28.8        28.8
##       Independent Comparisons     3433638     4639447
##                   OverAll r^2    0.042932    0.041995
##           Expected r^2 Sample    0.038558     0.03859
##                 Estimated Ne^        68.7        88.7
##             CI low Parametric        67.7        87.3
##            CI high Parametric        69.7        90.2
##              CI low JackKnife        37.1        48.5
##             CI high JackKnife       252.1       332.6
## 
## Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  /tmp/RtmpSfUVyo/dummy.gen/
## Completed: gl2genepop 
## $`2011_MBB`
##                     Statistic Frequency 1 Frequency 2
##  Lowest Allele Frequency Used       0.020          0+
##     Harmonic Mean Sample Size        40.5        40.4
##       Independent Comparisons     4137761     5352795
##                   OverAll r^2    0.027992    0.027775
##           Expected r^2 Sample    0.026634    0.026659
##                 Estimated Ne^       243.4       296.6
##             CI low Parametric       236.7         288
##            CI high Parametric       250.5       305.8
##              CI low JackKnife         130       160.1
##             CI high JackKnife      1272.5      1498.1
## 
## Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  /tmp/RtmpSfUVyo/dummy.gen/
## Completed: gl2genepop 
## $`2012_MBB`
##                     Statistic Frequency 1 Frequency 2
##  Lowest Allele Frequency Used       0.020          0+
##     Harmonic Mean Sample Size        51.5        51.5
##       Independent Comparisons     3691581     5663913
##                   OverAll r^2    0.022145    0.021786
##           Expected r^2 Sample    0.020591    0.020592
##                 Estimated Ne^       212.3       277.2
##             CI low Parametric         208       271.3
##            CI high Parametric       216.8       283.2
##              CI low JackKnife       128.1       166.7
##             CI high JackKnife       540.5         735
## 
## Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  /tmp/RtmpSfUVyo/dummy.gen/
## Completed: gl2genepop 
## $`2013_MBB`
##                     Statistic Frequency 1 Frequency 2
##  Lowest Allele Frequency Used       0.020          0+
##     Harmonic Mean Sample Size        63.3        63.2
##       Independent Comparisons     4167409     6045258
##                   OverAll r^2    0.018346    0.018056
##           Expected r^2 Sample    0.016576    0.016599
##                 Estimated Ne^       186.3       226.6
##             CI low Parametric       183.7       223.5
##            CI high Parametric         189       229.9
##              CI low JackKnife       126.3       153.3
##             CI high JackKnife       333.1         411
## 
## Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  /tmp/RtmpSfUVyo/dummy.gen/
## Completed: gl2genepop 
## $`2014_MBB`
##                     Statistic Frequency 1 Frequency 2
##  Lowest Allele Frequency Used       0.020          0+
##     Harmonic Mean Sample Size        22.9        22.9
##       Independent Comparisons     4454940     4454940
##                   OverAll r^2    0.051739    0.051739
##           Expected r^2 Sample     0.04988     0.04988
##                 Estimated Ne^       163.9       163.9
##             CI low Parametric       158.1       158.1
##            CI high Parametric       170.2       170.2
##              CI low JackKnife        75.9        75.9
##             CI high JackKnife         Inf         Inf
## 
## Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  /tmp/RtmpSfUVyo/dummy.gen/
## Completed: gl2genepop 
## $`2010_MBB`
##                     Statistic Frequency 1 Frequency 2
##  Lowest Allele Frequency Used       0.050          0+
##     Harmonic Mean Sample Size        28.9        28.8
##       Independent Comparisons     2469334     4639447
##                   OverAll r^2    0.043427    0.041995
##           Expected r^2 Sample    0.038533     0.03859
##                 Estimated Ne^        61.2        88.7
##             CI low Parametric        60.2        87.3
##            CI high Parametric        62.2        90.2
##              CI low JackKnife        31.9        48.5
##             CI high JackKnife       257.4       332.6
## 
## Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  /tmp/RtmpSfUVyo/dummy.gen/
## Completed: gl2genepop 
## $`2011_MBB`
##                     Statistic Frequency 1 Frequency 2
##  Lowest Allele Frequency Used       0.050          0+
##     Harmonic Mean Sample Size        40.5        40.4
##       Independent Comparisons     2167338     5352795
##                   OverAll r^2     0.02822    0.027775
##           Expected r^2 Sample    0.026616    0.026659
##                 Estimated Ne^       205.7       296.6
##             CI low Parametric         199         288
##            CI high Parametric       212.8       305.8
##              CI low JackKnife       109.8       160.1
##             CI high JackKnife      1023.9      1498.1
## 
## Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  /tmp/RtmpSfUVyo/dummy.gen/
## Completed: gl2genepop 
## $`2012_MBB`
##                     Statistic Frequency 1 Frequency 2
##  Lowest Allele Frequency Used       0.050          0+
##     Harmonic Mean Sample Size        51.6        51.5
##       Independent Comparisons     2092733     5663913
##                   OverAll r^2    0.022087    0.021786
##           Expected r^2 Sample     0.02055    0.020592
##                 Estimated Ne^       214.8       277.2
##             CI low Parametric       208.9       271.3
##            CI high Parametric       220.9       283.2
##              CI low JackKnife         134       166.7
##             CI high JackKnife       489.2         735
## 
## Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  /tmp/RtmpSfUVyo/dummy.gen/
## Completed: gl2genepop 
## $`2013_MBB`
##                     Statistic Frequency 1 Frequency 2
##  Lowest Allele Frequency Used       0.050          0+
##     Harmonic Mean Sample Size        63.4        63.2
##       Independent Comparisons     2091127     6045258
##                   OverAll r^2    0.018472    0.018056
##           Expected r^2 Sample    0.016556    0.016599
##                 Estimated Ne^       171.9       226.6
##             CI low Parametric       168.8       223.5
##            CI high Parametric       175.2       229.9
##              CI low JackKnife       117.3       153.3
##             CI high JackKnife       301.7         411
## 
## Starting gl2genepop 
##   Processing genlight object with SNP data
##   The genepop file is saved as:  /tmp/RtmpSfUVyo/dummy.gen/
## Completed: gl2genepop 
## $`2014_MBB`
##                     Statistic Frequency 1 Frequency 2
##  Lowest Allele Frequency Used       0.050          0+
##     Harmonic Mean Sample Size        22.9        22.9
##       Independent Comparisons     1950574     4454940
##                   OverAll r^2    0.052769    0.051739
##           Expected r^2 Sample    0.049838     0.04988
##                 Estimated Ne^       103.3       163.9
##             CI low Parametric        99.7       158.1
##            CI high Parametric       107.2       170.2
##              CI low JackKnife        48.8        75.9
##             CI high JackKnife     10445.4         Inf

Nb_LD_data_all <-  read_csv("./Results/NeEstimator/NeResultsSummary.csv")
Nb_LD_data_all$p_crit <- factor(Nb_LD_data_all$p_crit)

ggplot(Nb_LD_data_all, aes(x = Year, y = Nb_LD_est, colour = p_crit, group=p_crit)) + geom_point(position =dodge) + geom_errorbar(aes(ymin = Nb_LD_lower, ymax = Nb_LD_upper), width = width, position =dodge)  + coord_cartesian(ylim = c(0,1000)) + theme_bw()

Nb_LD_data <- Nb_LD_data_all %>% filter(p_crit == p_crit_chosen) #need to pick one(1) 0.05 is the original settings

write_csv(Nb_LD_data, "./Results/NeEstimator/NeResultsSummary_final.csv")

Cobmine and Save All Estimates Code

#read in from file so always works even if not a full rerun
Nb_LD_data <- read_csv( "./Results/NeEstimator/NeResultsSummary_final.csv")
Nb_LSA_data <- read_csv( "./Results/COLONY/ResultsSummary_final.csv")

#join and save
data <- left_join(Nb_LD_data, Nb_SA_data, by="Year")
write_csv(data, "./Results/ResultsSummary.csv")

Run Combination Estimates Code

data <- read_csv( "./Results/ResultsSummary.csv")

Ne_to_r <- function(Ne, S){
  r <- (-69*S^2 + sqrt(10000 * (S^4) * (Ne^2) + 4761 * (S^4) - 248400*(S^3)*Ne) + 1800*S*(Ne^2))/( 1800*(S^2)*(Ne^2))  
  return(r)
}

Ne_to_r_star <- function(Ne, S){
  r <- Ne_to_r(Ne, S)
  r <- r - 1/S
  return(r)
}


data$Var_SA_lower <- 0.25 * data$Nb_SA_est^4 * ((1 / data$Nb_SA_lower) - (1 / data$Nb_SA_est))^2
data$Var_SA_upper <- 0.25 * data$Nb_SA_est^4 * ((1 / data$Nb_SA_upper) - (1 / data$Nb_SA_est))^2
data$Var_SA_mean <- ( data$Var_SA_lower + data$Var_SA_upper ) / 2
data$Var_SA_weight <- 1 / data$Var_SA_mean


data$r_LD_lower <- Ne_to_r(data$Nb_LD_lower, data$S) 
data$r_LD_upper <- Ne_to_r(data$Nb_LD_upper, data$S) 
data$r_LD_est <- Ne_to_r(data$Nb_LD_est, data$S) 

data$rstar_LD_lower <- Ne_to_r_star(data$Nb_LD_lower, data$S) 
data$rstar_LD_upper <- Ne_to_r_star(data$Nb_LD_upper, data$S) 


data$Var_LD_lower <- (1/36) * data$r_star^-4 * (data$r_star - data$rstar_LD_lower)^2
data$Var_LD_upper <- (1/36) * data$r_star^-4 * (data$r_star - data$rstar_LD_upper)^2
data$Var_LD_mean <- ( data$Var_LD_lower + data$Var_LD_upper ) / 2
data$Var_LD_weight <- 1 / data$Var_LD_mean

data$weight_sum <- data$Var_LD_weight + data$Var_SA_weight

data$Var_LD_weightadj <- data$Var_LD_weight / data$weight_sum
data$Var_SA_weightadj <- data$Var_SA_weight / data$weight_sum

data$Nb_COM_est <- 1 / ((data$Var_SA_weightadj /data$Nb_SA_est) + (data$Var_LD_weightadj / data$Nb_LD_est))
data$var_COM <- 1 / ((1/data$Var_LD_mean) + (1/data$Var_SA_mean))

#Not CIs
data$Nb_COM_lower <- data$Nb_COM_est - sqrt(data$var_COM) 
data$Nb_COM_upper <- data$Nb_COM_est + sqrt(data$var_COM)

#nice table
data %>% dplyr::select(Year, starts_with("Nb_")) -> data_est_only

if(!include_2014){
  data_est_only <- data_est_only %>% filter(Year != 2014)
}else{
  data_est_only <- data_est_only
}

data_est_only %>% gt() %>% fmt_number(columns = c(5:10), decimals = 1) 

Year	Nb_LD_est	Nb_LD_lower	Nb_LD_upper	Nb_SA_est	Nb_SA_lower	Nb_SA_upper	Nb_COM_est	Nb_COM_lower	Nb_COM_upper
2010	61.2	31.9	257.4	96.0	58.0	194.0	74.7	54.9	94.5
2011	205.7	109.8	1023.9	234.0	144.0	492.0	222.5	169.2	275.9
2012	214.8	134.0	489.2	196.0	133.0	319.0	199.1	160.8	237.5
2013	171.9	117.3	301.7	152.0	109.0	221.0	154.2	128.9	179.5
2014	103.3	48.8	10445.4	127.0	71.0	358.0	114.3	81.4	147.2

Results

Outputs of NeEstimator and of Colony are saved in Report/NbResults.xlsx , where jacknife CIs are reported for pcrit = 0.05.

Summary Plots

Plot of Nb estimates with 95% confidence intervals

The combined estimates are also plotted here. The error bars in this case represent +- SD, not a confidence interval.

Plot of Combined Nb estimates +- SD

Stability

I take stability to mean that there are no significant differences between the Nb estimates for any years, this corresponds to the null hypothesis that Nb_2010 = Nb_2011 = Nb_2012 = Nb_2013.

LD

The situation is very difficult to evaluate graphically. The point estimates for 2010 and 2014 sit outside the common region (dark grey), but the 2011-2013 estimates (and the entirity of their CIs) lie entirely within the CIs for other years. No hard conclusions can be drawn from examining the individual CIs alone.

LD estimates with 95% CIs and values which fall in all CIs marked as a grey rectangle

In order to examine this formally, we will need to construct pairwise tests. This needs to be done with \(r^{*}\) as LDNe confidence intervals do not have normal CIs. The confidence intervals for \(r^{*}\) in LDNe are approximately normal. This method of comparison is statistically consistent with manner used in the construction of the CIs [cite: Jones, Ovenden and Wang (2016)].

A slightly more accurate result could be obtained by pulling the results directly from the LDNe code, and making minor adjustments to account for the fact this is only approximately normal, however I do not believe this would make a difference in this case.

The following table summarises the pairwise comparisons of the Nb estimates in terms of Nb. The first two columns identify which years are being compared, the third and forth columns state the relevant \(r^{*}\) values.

Their difference and the estimate for the standard deviation of the estimate follow. The combined standard deviation is found by taking the square root of the sum the of the two variances. This assumes both estimates are independent. While it is likely these estimates are correlated with each other, we have no way of accurately estimating their covariance so I believe it is best to assume independence.

The z-score and p-value are based on the ordinary z-test for the difference of two normaly distributed variables. The adjusted p-values are adjusted using the Bonferonni [actually used Holm FIX] method on the basis that we are making 6 pairwise comparisons and we wish to keep our family-wise error rate below 0.05. None of the unadjusted p-values were significant at the 0.05 level anyway.

data$rstar_LD_lower <- Ne_to_r_star(data$Nb_LD_lower, data$S) 
data$rstar_LD_upper <- Ne_to_r_star(data$Nb_LD_upper, data$S) 
data$rstar_LD_est <- Ne_to_r_star(data$Nb_LD_est, data$S) 
data$rstar_sd_mean <- (data$rstar_LD_lower - data$rstar_LD_upper) / (2*qnorm(0.9750))

results_frame <- data.frame("Year1" = NA, "Year2"= NA, "r1"= NA, "r2"= NA, "diff"= NA, "combined sd"= NA, "z-score"= NA, "p-value"= NA) %>% na.omit()

for(i in 1:(nrow(data)-1)){
  for(j in (i+1):nrow(data)){
    temp <- c(
      data$Year[i],
      data$Year[j],
      data$rstar_LD_est[i],
      data$rstar_LD_est[j],
      data$rstar_LD_est[i]-data$rstar_LD_est[j],
      sqrt(data$rstar_sd_mean[i]^2 + data$rstar_sd_mean[j]^2), 
      (data$rstar_LD_est[i]-data$rstar_LD_est[j]) / sqrt(data$rstar_sd_mean[i]^2 + data$rstar_sd_mean[j]^2),
      1-pnorm(abs(data$rstar_LD_est[i]-data$rstar_LD_est[j]) / sqrt(data$rstar_sd_mean[i]^2 + data$rstar_sd_mean[j]^2))
      
    )
    
    results_frame[nrow(results_frame) + 1,] <- temp
  
  }
}

results_frame$p.value.adj <- p.adjust(results_frame$p.value, "holm")
results_frame %>% gt() %>% opt_stylize() %>% fmt_number(columns = c(7:9), decimals = 3) %>% fmt_number(columns = c(3:6), n_sigfig =  3)

Year1	Year2	r1	r2	diff	combined.sd	z.score	p.value	p.value.adj
2010	2011	0.000891	0.000269	0.000622	0.000391	1.592	0.056	0.501
2010	2012	0.000891	0.000258	0.000634	0.000382	1.661	0.048	0.484
2010	2013	0.000891	0.000322	0.000570	0.000381	1.495	0.067	0.539
2010	2014	0.000891	0.000531	0.000360	0.000468	0.768	0.221	1.000
2011	2012	0.000269	0.000258	0.0000113	0.000137	0.082	0.467	1.000
2011	2013	0.000269	0.000322	−0.0000527	0.000135	−0.389	0.349	1.000
2011	2014	0.000269	0.000531	−0.000263	0.000304	−0.864	0.194	1.000
2012	2013	0.000258	0.000322	−0.0000640	0.000106	−0.607	0.272	1.000
2012	2014	0.000258	0.000531	−0.000274	0.000292	−0.938	0.174	1.000
2013	2014	0.000322	0.000531	−0.000210	0.000291	−0.721	0.235	1.000

SA

With the updated results, the intervals for the SA method are no longer mutually disjoint. Analysis is much the same as for LD now.

The analysis proceeds in a similar manner as for \(r^{*}\).

data$Nb_SA_est_inv <- 1 / (2*data$Nb_SA_est)

data$vstar_lower <- ((1/data$Nb_SA_lower - 1/data$Nb_SA_est) / (2*qnorm(0.9750)))^2
data$vstar_upper <- ((1/data$Nb_SA_est - 1/data$Nb_SA_upper) / (2*qnorm(0.9750)))^2

data$vstar_mean <- (data$vstar_upper+data$vstar_lower)/2

results_frame <- data.frame("Year1" = NA, "Year2"= NA, "2Nb^-1 1"= NA, "2Nb^-1 2"= NA, "diff"= NA, "combined sd"= NA, "z-score"= NA, "p-value"= NA) %>% na.omit()

for(i in 1:(nrow(data)-1)){
  for(j in (i+1):nrow(data)){
    temp <- c(
      data$Year[i],
      data$Year[j],
      data$Nb_SA_est_inv[i],
      data$Nb_SA_est_inv[j],
      data$Nb_SA_est_inv[i]-data$Nb_SA_est_inv[j],
      sqrt(data$vstar_mean[i] + data$vstar_mean[j]), 
      (data$Nb_SA_est_inv[i]-data$Nb_SA_est_inv[j]) / sqrt(data$vstar_mean[i] + data$vstar_mean[j]),
      1-pnorm(abs(data$Nb_SA_est_inv[i]-data$Nb_SA_est_inv[j]) / sqrt(data$vstar_mean[i] + data$vstar_mean[j]))
      
    )
    
    results_frame[nrow(results_frame) + 1,] <- temp
  
  }
}

results_frame$p.value.adj <- p.adjust(results_frame$p.value, "holm")

names(results_frame)[3:4] <-c('(2Ne)^-1 1', '(2Ne)^-1 2') 
results_frame %>% gt() %>% opt_stylize() %>% fmt_number(columns = c(6:9), decimals = 3) %>% fmt_number(columns = c(3:5), n_sigfig =  3)

Year1	Year2	(2Ne)^-1 1	(2Ne)^-1 2	diff	combined.sd	z.score	p.value	p.value.adj
2010	2011	0.00521	0.00214	0.00307	0.002	1.832	0.034	0.335
2010	2012	0.00521	0.00255	0.00266	0.002	1.608	0.054	0.486
2010	2013	0.00521	0.00329	0.00192	0.002	1.152	0.125	0.872
2010	2014	0.00521	0.00394	0.00127	0.002	0.599	0.275	0.920
2011	2012	0.00214	0.00255	−0.000414	0.001	−0.491	0.312	0.920
2011	2013	0.00214	0.00329	−0.00115	0.001	−1.329	0.092	0.735
2011	2014	0.00214	0.00394	−0.00180	0.002	−1.141	0.127	0.872
2012	2013	0.00255	0.00329	−0.000738	0.001	−0.901	0.184	0.920
2012	2014	0.00255	0.00394	−0.00139	0.002	−0.893	0.186	0.920
2013	2014	0.00329	0.00394	−0.000648	0.002	−0.414	0.340	0.920

After adjustment for multiple comparisons, there is still a significant difference between 2010 and 2011 (p = 0.045). The situation is somewhat confusing here as the estimates for 2011, 2012 and 2013 are not significantly different from 2010.

I had a number of reservations about this analysis, but I have included this analysis as I believe it is important.

These issues are:

• I am much less familiar with the CIs for COLONY than for LDNe
• The accuracy of extracting the variance from the CI bounds was very high, at the time of the original paper I put this down to the fact it rounds to whole numbers, but I’m not longer sure
• The estimate for 2010 has the narrowest CI bounds while having the lowest sample size. This is ‘correct’ and is based on the high number of sibling pairs assigned by COLONY.
• I have not been able to spend as much time checking this as I would like

Trends

I have also tested for (linear) trends. With only, five points the power to detect a trend is very low, but I think the exercise is worthwhile.

LD

ols <- lm(Nb_LD_est ~ Year, data=data_est_only)

lmfit <- broom::augment(ols, se_fit = TRUE, interval='confidence')

Using the method of Luikart et al 2020 (i.e. ordinary least squares regression on the Nb_LD estimates) there is no significant trend (p-value = 0.848). We can also see this graphically as the CI for the trendline (grey shaded area) includes the possibility of a flat line (i.e. slope=0, no trend).

gls3 <- nlme::gls(Nb_LD_est ~ Year, correlation=corCAR1(0.5, form = ~ 1, fixed=TRUE),  method='REML', verbose=TRUE, data=data)

# Design matrix for our observations ,, 
xmat <- model.matrix(~ Year, data=data)

# Regression coefficients
betahat <- coef(gls3)

glsfit <- broom.mixed::augment(gls3)

Sigmahat <- vcov(gls3)
#https://fw8051statistics4ecologists.netlify.app/gls.html
# var/cov(beta0 + beta1*X)
varcovhat<-xmat%*%Sigmahat%*%t(xmat)
SEline <- sqrt(diag(varcovhat))

glsfit$.upper <- glsfit$.fitted + qnorm(0.975) * SEline
glsfit$.lower <- glsfit$.fitted - qnorm(0.975) * SEline

Nor is there a trend when using a GLS regression that incorporates an estimate of autocorrelation (p-value = 0.808). This model only accounts for correlation and is not the only possible way to do this. It doesn’t resolve any of the other issues such as asymetric CIs, or incorporate other known information such as the variances of each estimate. However these tests for a trend are likely suitable here as 4 points are very unlikely to show evidence of a trend regardless.

SA

The same analysis can be repeated with the SA estimates.

ols <- lm(Nb_SA_est ~ Year, data=data_est_only)

lmfit <- broom::augment(ols, se_fit = TRUE, interval='confidence')

Using the method of Luikart et al 2020 (i.e. ordinary least squares regression on the Nb_LD estimates) there is no significant trend (p-value = 0.927).

gls3 <- nlme::gls(Nb_SA_est ~ Year, correlation=corCAR1(0.5, form = ~ 1, fixed=TRUE),  method='REML', verbose=TRUE, data=data)

# Design matrix for our observations ,, 
xmat <- model.matrix(~ Year, data=data)

# Regression coefficients
betahat <- coef(gls3)

glsfit <- broom.mixed::augment(gls3)

Sigmahat <- vcov(gls3)
#https://fw8051statistics4ecologists.netlify.app/gls.html
# var/cov(beta0 + beta1*X)
varcovhat<-xmat%*%Sigmahat%*%t(xmat)
SEline <- sqrt(diag(varcovhat))

glsfit$.upper <- glsfit$.fitted + qnorm(0.975) * SEline
glsfit$.lower <- glsfit$.fitted - qnorm(0.975) * SEline

Nor is there a trend when using a GLS regression that incorporates an estimate of autocorrelation (p-value = 0.896).

Discussion

It’s stable

Plot of Nb estimates with 95% confidence intervals

R Settings

R.version

##                _                           
## platform       x86_64-pc-linux-gnu         
## arch           x86_64                      
## os             linux-gnu                   
## system         x86_64, linux-gnu           
## status                                     
## major          4                           
## minor          3.3                         
## year           2024                        
## month          02                          
## day            29                          
## svn rev        86002                       
## language       R                           
## version.string R version 4.3.3 (2024-02-29)
## nickname       Angel Food Cake

References

Davenport, Danielle, Paul Butcher, Sara Andreotti, Conrad Matthee, Andrew Jones, and Jennifer Ovenden. 2021. “Effective Number of White Shark (Carcharodon Carcharias, Linnaeus) Breeders Is Stable over Four Successive Years in the Population Adjacent to Eastern Australia and New Zealand.” Ecology and Evolution 11 (1): 186–98.

Leberg, Paul. 2005. “Genetic Approaches for Estimating the Effective Size of Populations.” The Journal of Wildlife Management 69 (4): 1385–99.

Luikart, Gordon, Tiago Antao, Brian K. Hand, Clint C. Muhlfeld, Matthew C. Boyer, Ted Cosart, Brian Trethewey, Robert Al-Chockhachy, and Robin S. Waples. 2021. “Detecting Population Declines via Monitoring the Effective Number of Breeders (Nb).” Molecular Ecology Resources 21 (2): 379–93. https://doi.org/10.1111/1755-0998.13251.

O’Connor, J. 2011. “Age, Growth and Movement Signatures of the White Shark (Carcharodon Carcharias) in Southern Australia.” The University of Technology Sydney.

Wang, J. 2005. “Estimation of Effective Population Sizes from Data on Genetic Markers.” Philosophical Transactions of the Royal Society B: Biological Sciences 360 (1459): 1395–1409. https://doi.org/10.1098/rstb.2005.1682.

Waples, Robin S. 2022. “What Is N e, Anyway?” Edited by William Murphy. Journal of Heredity 113 (4): 371–79. https://doi.org/10.1093/jhered/esac023.